Sketch function from derivatives and limits

Q: Sketch function from derivatives and limits

This question evaluates understanding of calculus concepts—derivatives, concavity, limits, and continuity—and the ability to interpret derivative sign patterns to infer monotonicity and locate extrema.

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Statistics & Math questions require understanding of core concepts and practice. PracHub provides solutions with explanations to help you master statistics & math interviews.

Q: What difficulty level is this interview question?

This is a medium difficulty Statistics & Math question, commonly asked during Technical Screen rounds at Capital One.

Q: What role is this question designed for?

This question is commonly asked for Data Scientist candidates at Capital One during technical interviews.

Question

Function analysis with derivative and concavity constraints

Context (clarified): Assume f: R → R is differentiable everywhere and twice differentiable on R \ {2}. The second derivative f'' exists and has the stated signs on open intervals; f is continuous at x = 2 where f'' is undefined.

Given:

Limits: lim_{x→−∞} f(x) = 0 and lim_{x→+∞} f(x) = 5.
First derivative:
- f'(x) > 0 on (−∞, −2) and (1, 3).
- f'(x) < 0 on (−2, 1) and (3, ∞).
- f'(x) = 0 only at x ∈ {−2, 1, 3}.
Second derivative:
- f''(x) < 0 on (−∞, −1) and (−1, 2).
- f''(x) > 0 on (2, ∞).
- f'' is undefined at x = 2, but f is continuous everywhere.

Tasks: (a) Sketch a qualitatively correct graph of f, marking all local extrema and any inflection points. (b) Is it possible for f to have two local maxima and one local minimum consistent with the above? Justify rigorously. (c) Determine where an inflection point must occur, if any, and explain the sign of the slope immediately before and after x = 2.

Sketch function from derivatives and limits

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Sketch function from derivatives and limits

Function analysis with derivative and concavity constraints

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